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УДК 539.3
Коэффициенты уменьшения усталостной прочности на основе плотности энергии деформации для острых и тупых надрезов при многоосном нагружении
M. Abolghasemzadeh, Y. Alizadeh, H. Mohammadi
Технологический университет им. Амира Кабира, Тегеран, 15875-4413, Иран
При эксплуатации механизмов возникают условия для зарождения и распространения усталостных трещин в местах концентрации напряжений, обусловленных геометрическими особенностями деталей, такими как надрезы, углы, отверстия, кромки сварного шва и т.д. Классический анализ усталости образцов с надрезом проводят на основе эмпирической формулы и подобранного коэффициента уменьшения усталостной прочности. При этом такой анализ требует больших затрат и не в полной мере обоснован физически. В работе представлена простая и эффективная методика оценки усталости деталей с надрезом при многоосном нагружении. В рамках метода на основе точного определения объема материала конечного размера вокруг очага усталостной трещины (вершины надреза) и усреднения энергии деформации по данному объему исследовано влияние на морфологию зоны предразрушения различных режимов деформации (I, II, III) и коэффициента нагрузки. Для деталей с острым или тупым надрезом достаточно провести анализ линейно упругой конечно-элементной модели и иметь данные о свойствах материалов, полученные в ходе простых одноосных испытаний. Получены новые соотношения для определения действующего напряжения (растяжения) и коэффициента снижения усталостной прочности для образцов с надрезом. Точность предложенной модели подтверждена на основе имеющихся экспериментальных данных для цилиндрических образцов с острыми и тупыми надрезами при изгибном кручении. Такие условия часто встречаются в механизмах, применяемых в прокладке трубопроводов, в автомобилестроении, энергопроизводящих установках, при бурении и др.
Ключевые слова: предел усталости, образцы с U и V надрезом, конечно-элементная модель, зона процесса
DOI 10.24411/1683-805X-2019-11009
Fatigue strength reduction factors based on strain energy density applied to sharp and blunt notches under multiaxial loading
M. Abolghasemzadeh, Y. Alizadeh, and H. Mohammadi
Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, 15875-4413, Iran
Real mechanical assemblies favor the initiation and propagation of fatigue cracks due to stress concentration phenomena arising from the geometrical features such as notches, corners, holes, welding toes, etc. Classical fatigue analysis of notched specimens is done using an empirical formula and a fitted fatigue strength reduction factor, which is experimentally expensive and lacks physical scene. In the present paper, a simple and meaningful methodology is proposed to assess notched components against multiaxial fatigue. In this method, by precisely defining a finite-size volume surrounding the fatigue crack initiation site (notch tip), over which the strain energy is averaged, the morphological effect on the process zone is fully addressed. Such a method takes into account the effect of combination of different modes (I, II, III) and the load ratio. In order to implement it for components with a sharp or blunt notch, it is enough to analyze a linear elastic finite element model and to know only the properties of materials obtained from simple uniaxial tests. New relationships for determining an effective (tensile-type) stress and fatigue strength reduction factors are derived for notched specimens. The accuracy of the proposed model is validated by experimental data available in the literature, related to tubular specimens weakened with sharp/blunt notches under combined bending-torsion loading. Such a situation widely appears in equipment used for various branches of industry such as piping, automotive, power plant, drilling, etc.
Keywords: fatigue strength, U and V notched specimens, finite element model, process zone
1. Introduction
Structural engineers engaged in fatigue analysis are always concerned with stress concentration phenomena due to their detrimental effect on the fatigue strength and life of
the mechanical components. Significant efforts have been made to formulate the stress field in the vicinity of a crack [1] and a notch tip [2]. Additionally, three dimensional effects on the stress field have been investigated [3-5]. In
© Abolghasemzadeh M., Alizadeh Y., Mohammadi H., 2019
conventional fatigue analysis, a knock-down factor or so-called fatigue notch factor Kf is considered to take into account the detrimental effect of stress concentration on the S-N curve which is defined by the ratio of unnotched (smooth) fatigue strength to notched fatigue strength. From a simplistic point of view, smooth and notched components would be expected to have the same fatigue life in case the stress in the smooth component is equal to the stress Kt S at the notch in the notched component, where Kt is the stress concentration factor and S is the remotely nominal stress, i.e., Kf = Kt. However, experimental data show that Kf may be less than Kt. This has led to introducing the socalled notch sensitivity factor as q = (Kf -1)/(Kt -1), where q varies between 0 and 1 [6].
There is a consensus amongst fatigue researchers that the Kf < Kt effect is related to the stress gradient near the stress concentration site. One of the arguments based on stress gradients states that the material is not sensitive to the peak stress, but rather to the average stress that acts over a small, but finite, size region. In other words, some finite volume of material is involved in the initiation and continuation of the fatigue damage process. This active region is well known as the process zone. Another effect of the stress gradient can be explained according to statistics. Note that the fatigue damage process may initiate in a crystal grain that has an unfavorable orientation of its slip planes in relation to the planes experiencing maximum shear stress, or in other cases at a void, inclusion, or other microscopic stress raisers. Many potential damage initiation regions appear within the volume of a smooth specimen. However, at a sharp notch, it is possible that no such damage initiation region appears in the small region in which the stress is near its peak value. Therefore, on the average, compared to what is expected, the notched component can have a greater fatigue strength if the comparison is made based on the local notch stress KtS [7].
Although in practical applications many structures and components are subjected to multiaxial cyclic loading conditions, this is still true for the ones having geometrical features under nominal uniaxial cyclic loading, because the stress fields acting on the fatigue process zone are always at least biaxial. Moreover, it is evident that, for a given cyclic stress field damaging the process zone, the fatigue strength has to be the same independently of the source from which the multiaxiality of the stress field itself arises. This argument is the reason why modern methods used to evaluate fatigue of structures with geometrical features (such as holes, notches, and edges), and in particular welding joints, are mainly based on local quantities. The main aim of developing these methods is to modify the effect of the stress gradient existing in the stressed area on the fatigue strength of the structure. The pioneering study of the notch effect on fatigue strength goes back to the studies of Neuber and Peterson. From a design perspective, Neuber proposed
to calculate the effective stress by averaging the linear elastic stress near the notch tip over a microstructural unit of the material (structural particles or crystals), rather than the stress at a singular point. Paterson suggested that the effective stress can be calculated directly using the value of the linear elastic stress at a given distance from the notch tip. These assumptions are empirical and lack physical foundation. The material parameters used in these approaches need to be calibrated using extensive experimental data of various notched specimens with different notch geometries. The common feature of the mentioned models is that the effective stress depends on the material characteristic length, which is basic to the theory of critical distance [8].
A drawback of Neuber's and Peterson's models is the assumption that the notch tip radius is the only geometrical parameter governing the distribution of elastic stress in the fatigue process area. It has been recognized that the notch opening angle is also an important parameter for evaluating the linear elastic stress field (especially for angles larger than 90°). Therefore, the use of the notch stress intensity factor in determining the stress field around the notch, which takes into account the notch opening angle, was suggested for summarizing the fatigue results in a single scatter band
[9-11].
Since the implementation of the notch stress intensity approach for evaluating the fatigue of notched specimens with different angles leads to different results, the strain energy density criterion is proposed based on the exact definition of the finite-size structural volume around the notch (over which the strain energy is averaged) [12, 13]. Compared to the notch stress intensity factor, the advantage of the strain energy density criterion is that the critical value of the strain energy density is independent of the notch geometry. Berto et al. [14] provided a large bulk of fatigue data for V-notched and semicircular notched specimens made of 40CrMoV13.9 steel under pure tension, pure torsion, and multiaxial loading. Analyzing the results in terms of the mean value of the strain energy density over a finite-size control volume, they concluded that for 40CrMoV13.9 steel the radius of the control volume is independent of the loading mode.
Apart from fatigue, the strain energy density criterion was extensively validated for fracture assessment of notched components with different geometries [15-18]. It was employed to investigate fracture and fatigue of notched specimens made of different types of materials such as graphite [15, 19], polymethylmethacrylate [20, 21], steels [22, 23], polyurethane [24, 25], titanium alloys [26-28], tungsten-copper functionally graded materials [29-33], and silicon [34]. The strain energy density criterion was also used to assess the high temperature fatigue of notched components made of different advanced materials [35, 36].
It should be noted that common fatigue experiments are performed using uniaxial loading. Therefore, the main
feature of multiaxial failure criteria is to quantify an equivalent (tensile-type) stress, which can then be compared with the uniaxial mechanical properties of a metal. The equivalent tensile stress can be either scalar or vector. The scalar case is the development of static hypotheses of material resistance to fatigue by replacing static stress values with amplitudes or ranges of dynamic loading. The Von Mises and Tresca criteria are examples of the scalar case. In the vector case, the critical plane is defined normal to the maximum value of the vector, on which failure is predicted to occur. Numerous approaches have been proposed so far based on the critical plane concept, the most popular of which are Fatemi-Socie and Smith-Watson-Topper for shear- and tensile-mode materials, respectively. A comprehensive review of the critical plane methods is available in the literature [37-39], and therefore they are not considered here.
In the present paper, based on the averaged strain energy density criterion attempts are made to derive relationships for determining an effective (tensile-type) stress and fatigue strength reduction factors for notched specimens. In this method, by precisely defining the structural volume, the morphological effect on the process zone is fully addressed. The method takes into account the effect of combining different modes (I, II, III) and the load ratio. In order to implement the method for sharp/blunt notched components, it is enough to run a linear elastic finite element model and to know only the properties of materials obtained from simple uniaxial tests. The accuracy of the proposed model is validated by experimental data from the literature.
2. A brief review of the fatigue cracking mechanism in metals
From an engineering perspective, material cracking behavior under cyclic loading can be investigated at the micro-, meso-, or macroscale. To be precise, a microcrack is located within a single grain, a mesocrack covers several grains, and finally a crack that includes a large number of broken grains is called a macrocrack. When a metal or an assembly of grains (crystals) is subjected a state of stress/ strain, lattice imperfections (dislocations) can move due to the shearing forces acting on certain atomic planes resulting in the formation of thin lamellae, which are called persistent slip bands. Owing to the presence of persistent slip bands, an alternation of extrusions and intrusions appears on the grain profile. After a certain number of cycles, microcracks initiate in the grains either due to microstress concentration phenomena resulting from the presence of deep intrusions or due to separation occurring within the slip bands. A microcrack can become a mesocrack only if the applied cyclic loads are capable of forcing the crack to break through the microstructural barriers located beside the grain in which the microcrack initiated. As the crack
grows, the stress intensity increases at the crack tip and the crack accelerates, while the microstructure effect gradually decreases. When the above cracks reach a length of the order of a few grains, the initial mode II dominated crack propagation ends and the cracks tend to orient themselves so that to experience the maximum mode I loading. As a result, crack growth can slowly change from stage I (shear crack) to stage II (tensile crack). Because of the 3D nature of the crack path, the major part of the metal fatigue strength is related to the crack growth region after the boundary of the first grain to a depth of 5 to 10 grains. All the above mentioned physical processes, in the presence of a notch, are rapidly diverted towards the growth of stage II due to the initial growth of the crack in the plastic region around the notch [8, 40].
It can be hypothesized based on the above discussed physical processes that fatigue damage of uncracked (smooth or notched) bodies can successfully be predicted by modeling the initiation and initial propagation of micro/meso-cracks. The only way to correctly model the micro/me-socrack behavior is to consider the real material morphology and the elastic-plastic behavior of grains. Unfortunately, the estimation of fatigue strength by rigorously considering all the aspects mentioned above, even if it is definitively attractive from a theoretical point of view, would be too difficult to perform for the assessment of real components.
In general, in the case of components weakened by a notch of nonzero radius under plane stress-strain conditions, the relationship between stresses and elastic-plastic strains at the notch tip in the local plastic region is approximated by Neuber's law. Neuber's law states that the total strain energy density (summation of volumetric and de-viatoric strain energy densities) at the notch tip is equal to such a quantity for the condition that the material with elastic behavior and the same geometry is subjected to the same loading. It should be noted that Neuber's law was initially developed for notched components under pure shear. This means that the Neuber equation only represents the equality between the deviatoric strain energy densities. Accordingly, in this paper, the deviatoric part of the strain energy density function is used to evaluate the fatigue of notched components under bending and torsional loading. In order to develop Neuber's law for sharp notches, Lazzarin and Zambardi suggested that the strain energy concentration evaluated for a finite-size structural volume around the notch is independent of the structural model of the material [41]. According to their hypothesis, when the localized yielding conditions are met, the strain energy calculated by considering either elastic or plastic behavior leads to the same result. The validity of this hypothesis was examined with finite element results for two cases of localized yielding and large-scale yielding. It follows from the hypothesis that by directly post-processing the results of simple linear elas-
tic finite element analysis the strain energy density criterion has the ability to address the fatigue crack growth process in the presence of plasticity at the notch tip, provided that the nominal stress on the structure is less than the yield stress of materials.
In addition, choosing the strain energy density parameter as the driving force of the fatigue process, instead of the stress or strain parameters alone, eliminates the complexities associated with very fine mesh or the use of special elements around the stress concentration site in determining the exact stress and strain components, which reduces computational and time costs.
The criterion used in this article will be described in detail in the next section.
3. Strain energy density criterion for multiaxial fatigue problems
The strain energy density criterion over a finite volume has been based on an accurate definition of the structural volume and on the fact that the critical energy is independent of notch sharpness. Based on the strain energy density criterion, fracture occurs when the change of the strain energy averaged over the structural volume AW reaches a critical value A Wc, which is a material parameter [12]. In the case of sharp notches, the strain energy density values are determined using closed-form equations in terms of the notch stress intensity factor values for different loading modes. As for blunt notches, some equations are derived to relate the strain energy density values of notched specimens under bending and torsion to the maximum principal stress and the maximum shear stress of the notch tip, respectively. The next sections describe equations for the fatigue strength reduction factors of sharp and blunt notches.
3.1. Implementation for components weakened by a sharp notch
In the notch stress intensity approach, which was initially proposed for welded joints, the weld toe is assumed to be a notch with a zero tip radius and the fatigue behavior is associated with the asymptotic distribution of stress in the vicinity of the notch tip. Recent experiments on cruciform joints have shown that most of the lifetime of such welded joints is spent during crack nucleation and subsequent propagation within a high-stress area near the weld root (a region described only by the notch stress intensity factor). In the shadow of this fact, the notch stress intensity factor can be regarded as a quantity for estimating the fatigue life of welded joints. By defining the cylindrical coordinates (r, 6, z) at the notch tip (Fig. 1), the linear elastic stress distributions are expressed in terms of the notch stress intensity factor values as follows [10]:
M t
model: a® = Kf fj (e)Z1-1, mode II: af = Kjj gtJ (e) -1, mode III: j = K:nn hlj(e)r^-1,
(1)
Fig. 1. Local coordinate system for a notched specimen under multiaxial loading (color online)
and
mode I: sin(2X1y) = -X1sin(2y), mode II: sin(2X2y) = +A2sin(2y), (2)
mode III: X3 =n/2y, where ÀI; An, and XIII are the eigenvalues of the elastic problem, which can be obtained from solving Eqs. (2). The parameters Kj1, Kjj, and Kl are the notch stress intensity factors for stress distributions of mode I, mode II, and mode III, respectively, which are defined as follows [8] : K1 = lim rV1 aee ( r, e = 0),
r^0+
K1 = V2n lim rÀ2-1 are (r, e = 0),
r^0+
(3)
K1 = V2n lim Z3"1 aez(r, e = 0).
r^0+
The functions f, gj, and hj are defined as [8]: mode I:
frr
fre
1
1
V2n (1+X1)+X1(1 -X0
(1+X1)cos(1 -X1) e" (3-X1)cos(1 -X1)e (1 -X1)sin (1 -X1)e
= -sin(1 -X1)e X1 sin(1+X1)e, mode II:
+X1 (1-X1)
cos (1+X1)e " - cos(1+X1)e sin (1+x1 )e
(4)
grr
gre
1
V2n (1 -X2)+X2(1+X2)'
-(1+x2)sin(1 -x2) e" -(3 -X2)sin (1-x2)e (1 -X2)cos(1 -x2) e
+X2 (1+X )
-sin (1+x2) e sin (1+x2 )e cos (1+X2)e
X
In the case of sharp notches, the structural volume of the strain energy density criterion is considered to be a circular sector around the notch tip, the radius of which Rc is a material property independent of the notch geometry and loading mode.
For a structural volume with the radius Rc as illustrated in Fig. 2, the components of the deviatoric strain energy density in the volume are obtained as follows: ed
mode I: AWd1 = (AKtn )2 Rc2(V1), E
Notch bisector
Fig. 2. Structural volume in the strain energy density criterion sin (1 -A,2 )6
X2 = - sin (1+A2 )e' mode III:
mode II: AWd2 =
— (akU )2 R2( À2 -1),
E ed
(6)
mode III: A Wd3 = ed3 (AK1 )2 R2(-1) E
The eàj coefficients (j = 1, 2, 3) are the functions of the notch opening angle 2a and Poisson's ratio v according to the following relationships:
1 + -7.
I [/rr + Zee + fzz ( frrfee +
1
sin X36
6À1Y-.
+ fee /rr + /rr/zz ) + 3 /re]de,
V2n [cosx3e
In many practical applications, the notch stress intensity factors can be linked to the nominal stresses according to the following expressions [9, 10]:
ed2 =
1 + V?[ 2
J [ grr
+ g ee + glz - ( grrg ee +
(7)
AKJ1 = k1 Aa0 t AKn = k3 At0 t
1-À,
AKH = k2 Aa0 t
,1-à,
1-à3
(5)
where t is the characteristic dimension (e.g., thickness), k1, k2, k3 are the nondimensional coefficients dependent on the structure geometry (similar to the stress concentration factor), and Aa0, At0 are respectively the nominal tensile and shear stresses at a distance far from the notch singularity.
6^2 Y-y
+ g66 grr + Srr&zz ) + 3 gr26]d6,
1 + -+Y 2 2 ed3 = TT-I + hl)]d6.
In the above equations, y is equal to n - a (Fig. 2). Since the examples provided in this paper are given to validate the model under dominant loading modes I and III, only the values ed1 and ed3 are presented in Table 1 for different values of the notch opening angle and Poisson's ratio.
In pure mode I, pure mode III, and mixed mode I + III loading conditions, the critical conditions of fatigue accord-
Table 1
The values of ed1 and ed3 calculated using the deviatoric strain energy density for different conditions
Plane strain/stress ed3 Plane strain ed1 Plane stress ed1 2a
V = 0.3 V = 0.4 V = 0.34 V = 0.32 V = 0.3 V = 0.25 V = 0.3
0.4138 0.058675 0.0606 0.0616 0.0628 0.066315 0.1207 0°
0.3966 0.061613 0.0636 0.0646 0.0658 0.069417 0.1257 15°
0.3793 0.065310 0..0672 0.0682 0.0694 0.072968 0.1303 30°
0.3621 0.069423 0.0711 0.0720 0.0732 0.076616 0.1340 45°
0.3448 0.073564 0.0750 0.0758 0.0768 0.079967 0.1362 60°
0.3276 0.077265 0.0783 0.0790 0.0798 0.082571 0.1363 75°
0.3104 0.079993 0.0806 0.0811 0.0817 0.083948 0.1338 90°
0.2931 0.081208 0.0813 0.0816 0.0820 0.083659 0.1284 105°
0.2759 0.080463 0.0801 0.0802 0.0804 0.081400 0.1202 120°
0.2586 0.077512 0.0767 0.0766 0.0767 0.077084 0.1094 135°
0.2414 0.072361 0.0712 0.0710 0.0709 0.070848 0.0969 150°
ing to the strain energy density criteria read
mode I: Q ^(AKf )2Rc2(Xi-1) < AWc, E
mode III: Cm^(AK^)2Rc2(À3-1) <AWc, E
mixed mode I + III: CI ed1 (AK* )z RAAl~lj + E
+ Cni%AKnt)2R2(X3-1) <AWc,
E
where the coefficients CI and CIII take into account the load ratio RL = Gmin/Gmax as follows:
)2 r>2(Ài -1)
(8)
C
I/III
_Awl =
A Wo
1 + R
(1 - Rl) 1 - R2
V -1 < Rl < 0,
(9)
(1 - Rl)2
-, 0 < Rl < 1,
in which AWL and AW0 are defined schematically in Fig. 3. In particular, for RL = -1 and 0 the values of the coefficients are 0 and 0.5, respectively.
In addition, the critical value of the change of the de-viatoric strain energy density is
AW = C
1 + v
0
Ac.
(10)
In the above relation, Ag^ is the fatigue limit corresponding to a reference number of cycles under uniaxial cyclic loading with the corresponding coefficient C0 used to account for load. Therefore, by taking into account the critical value of the notch stress intensity factor under mode I loading condition AK^, the size of the structural volume corresponding to the same reference number of cycles is defined as [10]:
1
Rc =
3
1 + v
ed1
LI,th
ACT
2(1-^1)
(11)
It should be noted that for the use of Eq. (11) the values of AK *th and Ag^ obtained from the test in the same load ratio (CI = C0) must be substituted.
By substituting AWc from Eq. (10) and the notch stress intensity factors from Eq. (5) in Eq. (83), we come to the following relation for defining the tensile nominal effective stress:
Co AC f = Ci( K CAC o)2 + C,
Iii
AC. At
K fTAT o
KfC =
1 + v
ed1 ¿1
t
Rc
(12)
AT.
AcJ 1 + v
N1-^3
-ed3 k3
Rc
where KfG and Kf are the notch fatigue strength reduction factors under bending and torsion, respectively. For an unnotched specimen, the values of these factors are equal to unity. Finally, Eq. (83) can be simplified as follows: AGeff <AG^ . (13)
It is worthwhile mentioning that for an unnotched specimen, according to Eq. (121), the effective stress turns out to be the von Mises equivalent stress by substituting =
because both are extracted from the deviatoric strain energy density. Therefore, the proposed criterion can be interpreted as a developed form of the von Mises criterion for notched components.
From a practical point of view, for a specific geometry and static state loading conditions, the averaged value of strain energy in a control volume of arbitrary radius R is calculated as expressed in the strain energy density criterion. By combining Eqs. (5), (122), (123), the corresponding fatigue strength reduction factors are determined from the following equations:
KfC =
KfT =
' R ^
Rc
c
At.
Ac.
R
Rc
1 + v
1 C0
EWi— ,
/
(14)
1 + v
-EW,
III
The far-field stresses g0 and t0 are calculated based on the static load applied in the finite element model.
Fig. 3. Effect of the load ratio RL on the strain energy density change during a loading cycle. Rle [-1, 0] (a), Rle [0, 1] (b)
3.2. Implementation for components weakened by a blunt notch
In the case of blunt notches, the structural volume assumes a crescent shape, with Rc being its maximum width as measured along the notch bisector line. By considering the coordinate system of which the origin is located at a distance r0 from the notch tip (r0 depends both on the notch root radius p and the opening angle 2a, according to the expression r0 = p[n- 2a]/[2n - 2a]), the mean value
of the strain energy density can be expressed as
( R ^
W = F(2a)H - c
2a, v,—— P
(15)
where the values of F and H are reported in Ref. [13] as the functions of the opening angles, Poisson's ratios, and the ratio of Rc to the notch tip radius.
On the other hand, by considering a local system that is the same as the previous one, Zappalorto et al. [42] have recently developed an analytical solution for the local stress fields induced by circumferential U- and blunt V-shaped notches in solid bars under torsion. According to this solution, Lazzarin et al. [43] developed expressions for the strain energy density over a structural volume which for U- and blunt V-shaped notches under torsion read
' 2a, R
Wm = W3
V
max
2G
(16)
and the values of w3 are
v J where G is the shear modulus reported in Ref. [43] as functions of the notch opening angles and the ratio of Rc to the notch tip radius.
Using the definition of the fatigue strength reduction factors, in combination with Eqs. (15), (16), the fatigue notch factors for a blunt notch under bending and torsion can be obtained as follows:
Kp,f =
FHKb , Kp f =
At
■^3w3 K2, (17)
11 + v ^ Act^
where Kb and Kt are the stress concentration factors of bending and torsion, respectively, which are equal to the ratio of the maximum stress at the notch tip to the nominal stress.
3.3. A more precise explanation about the size of the structural volume
The socalled fatigue limit can be defined in terms of the formation of micro/meso nonpropagating cracks that initiate in the weakest points of components. In smooth specimens, or in the presence of very blunt notches, the propagation of such cracks is arrested either by the first microstructural barrier or by the first grain boundary (Fig. 4, a). On the contrary, in the presence of sharp notches, nonpropagating cracks are seen to be much longer and their length depends on the morphology as well as on the fatigue properties of the considered material (Fig. 4, b). An intermediate situation can be thought for blunt notches. Gener-
ally speaking, in the case of notched components, the stress concentration factor is a measure for fatigue crack initiation, but in order to continue the crack growth further to the singularity area, the crack length must reach a critical value corresponding to the threshold value of the stress intensity factor AKth.
According to the model proposed by Yates and Brown [44], the maximum nonpropagating crack length is equal to 1/ n (AKth/( FAct^ ))2, where F is a geometrical factor whose value is unity for an infinite plate with a central crack subj ected to fully reversed tension-compression cyclic loading conditions (this configuration is assumed to be representative of the pure material cracking behavior as geometry does not influence the crack growth), and less than unity for other cracked situations. Therefore, in general, the maximum crack propagation length governed by the microstructure can be considered to be equal to (£NPC)max = = 1 n (AKth/Act^ )2. This value is equal to the material characteristic length L in the theory of critical distance. Now, the relationship between the size of the structural volume
Ac.
Crack propagation in smooth bodies under fatigue limit condition
Crack propagation in notched bodies under fatigue limit condition
Ideal crack propagation under fatigue limit condition
0
Nonpropagating crack
0
Acc
Structural volume in SED approach
Fig. 4. Schematic description of the cracking behavior in smooth (a), notched bodies (b), and definition of the material structural volume (c), SED — strain energy density (color online)
in the strain energy density criterion Rc and (¿NPC)max can be interesting. According to Eq. (11), the value of Rc under plane strain condition is calculated to be about half of (ZNPC )max. In parallel, the structural volume of a cracked component is a circle of radius Rc centered on the crack tip (see Fig. 4, c).
Referring to Fig. 4, c again, Rc can be interpreted as the longest ideal crack, the propagation driving force of which is affected by the microstructure of the material. With this argument, it can be concluded that in the strain energy density criterion the effect of microstructure on the nucle-ation and initial growth of a micro/meso fatigue crack is fully addressed by the exact definition of the structural volume. To put it more practically, independent of the geometrical features (smooth, sharp, or blunt notch), the fatigue cracking mechanism is controlled by the uniaxial fatigue limit of the material.
4. Validation of the proposed model
The accuracy of the proposed model in predicting the fatigue limit of notched components was assessed with the experimental data available in the literature obtained by testing notched specimens under pure bending, pure torsion, and combined bending-torsion. Sharp and blunt notches were considered. Most of the multiaxial loading was in-phase, but there are several cases of out-of-phase loading. In particular, the fatigue limit of a tube-to-flange connection, which has industrial application and was obtained experimentally under combined bending-torsion loading, was compared with the results of the proposed model.
In all cases, the error index is calculated as follows:
Error =A°eff Aa~xi00, Aa
(18)
4.1. Application for a welded tube-to-flange joint (2a = 135°)
The first set of data used for the validation of the method refers to tubular welded joints made of Fe E 460 with an ultimate tensile strength of 670 MPa. The typical application of this connection is in areas with the highest stresses in hot-blast furnaces where the connection is subjected to bending and torsion with constant amplitude and with con-stant/nonconstant principal stress direction.
Fatigue tests were carried out on the mentioned joint under fully-reversed loading (R = -1), pure bending, pure
torsion, and combined bending-torsion with the ratio of the shear stress range to the stress range equal to 0.58 [45]. The results of the tests are presented in Table 2.
The value of 440 MPa is reported for the fatigue limit of tube-to-tube joints made of Fe E 460 with ground weld beads under pure bending. So, by using the value pertaining to the tube-to-flange joints A^n(j2a=135) = 86.9 MPam0326 together with v = 0.3 and edl = 0.0766, Eq. (11) results in Rc = 0.4896 mm under the hypothesis of plain strain conditions.
Commercial software ABAQUS was used for the finite element analysis of the joint. The analyses were performed under the linear elastic hypothesis. The section view of the model, accompanied by the assumptions made in the software, is depicted in Fig. 5, a. The weld toe radius is assumed to be equal to zero (sharp notch). This assumption is an engineering choice because it considerably reduces the number of local geometrical configuration. The tubular part of the connection has an outer diameter of 88.9 mm and an internal diameter of 68.9 mm. The sheet has a thickness of 25 mm and a diameter of250 mm. In order to accurately determine the strain energy at the vicinity of the weld toe, by dividing the model into a number of partitions, a refined mesh was generated in areas near the notch and a relatively coarse mesh was generated at distances away from it (see Fig. 5, b).
With respect to geometry and loading, all three loading modes are induced around the notch. Additionally, it has been shown that the contribution of the mode II stress components, when the notch opening angle is larger than 102°, is not singular and therefore the fatigue strength of the notched specimen can be estimated with high accuracy in terms of the stress intensity coefficients of modes I and III. Assuming that the loading modes are independent, two analyses were carried out applying the bending moment in one model and the torque in another one. To apply the load in ABAQUS, the whole surface of the free edge is coupled to a reference point and remote loads (bending moment for mode I or torque for mode III) are applied to this point (see Fig. 5, a). Note that due to symmetry only one half of the geometry was considered for bending loading. In order to use Eqs. (14) for determining the strength reduction factors, a circular sector with the radius R = 1 mm was considered around the notch tip. 8-node brick elements with reduced integration (1 integration point) and a linear shape function (C3D8R) were used in the control volume and the
Table 2
Predicting fatigue limits of tube-to-flange joints of Fe E 460 subjected to bending and torsion
Material Loading o0, MPa t0, MPa Fatigue limit, MPa Oeff, Eq. (121), MPa
Bending 218 - 447.7
Fe E 460 steel Torsion - 172 440 449.5
Bending+torsion 174 101 444.2
Fixed boundary condition
Weld toe radius is set equal to zero (sharp notch)
Coupling-type
Remotely applied loads T
M
Boundary of control volume b
K/
Fig. 5. Geometry of the welded joints (a), position of the control volume (b), meshed model (c), RP — reference point (color online)
areas around it. Figure 5, c illustrates the mesh generated in the control volume. It is important to remember here that, thanks to several recent papers, it has been demonstrated that the value of the strain energy density is not sensitive to the degree of mesh refinement [30, 46-49].
The averaged strain energy density over the control volume was extracted from the finite element analyses. The
results of the linear elastic analyses in terms of the strain energy density value over the control volume with R = 1 mm are reported in Table 3. Here all the results are related to a nominal load equal to 1000 Nm, Young's modulus E = = 207 GPa, and Poisson's ratio v = 0.3. Finally, the fatigue effective stress of the joint under different loading conditions was determined using Eqs. (14) as given in Table 2.
Table 3
The values of strain energy density and fatigue strength reduction factors as determined from finite element (FE) analysis
Mode FE simulation Fatigue strength reduction factors
R, mm SED, N/mm2 KT, Eq. (14j) KT (Ao„/At„ ), Eq. (142)
I 1 2.85 x10-15 2.0535 -
III 1 1.38X10"15 - 2.6136
Table 4
Experimental data from Ref. [40] under multiaxial fatigue loadings
Material Fatigue limit, MPa L, mm p, mm Kb Kt Notch type
0.4% C steel (normalized) 332 0.178 0.005 18 8.6 Sharp
3% Ni steel 343 0.144 0.005 18 8.6 Sharp
3/3.5% Ni steel 352 0.516 0.01 13.3 6.7 Sharp
Cr-Va steel 429 0.101 0.011 12.1 6.2 Sharp
3.5% NiCr steel (normal impact) 540 0.150 0.022 8.7 4.7 Sharp
3.5% NiCr steel (low impact) 509 0.109 0.022 8.7 4.7 Sharp
NiCrMo steel 594 0.106 0.031 7.5 4.2 Sharp
SAE 1045 304 0.159 5 1.6 1.3 Blunt
S65 584 0.056 0.838 1.5 1.2 Blunt
Accuracy in predicting fatigue limits of sharp V-specimens (2a = 60°) under bending and torsion
Table 5 ■notched
(^L=-1)
60°,
Material o0, MPa T0, MPa Fatigue limit, MPa oeff, MPa
0.4% C steel 76.7 140 332 393.7
120.1 107.3 347.3
148.4 74.7 314.6
166.1 51.7 305.6
180.5 27.6 305.4
3%Ni steel 72.6 132.8 3 407.2
126.7 106.4 385.3
171.5 83.5 395. 2
195.6 56.8 392.4
209.2 25.3 387.9
3/3.5%Ni steel 97.6 172.6 352 315.2
175.5 153.4 317.3
214.4 108.4 281.9
265.4 78.7 293.3
286.2 40.1 288.2
Cr-Va steel 77.3 147.3 429 520.0
137.5 117 491.5
172 88.6 475.8
210.3 58 494.8
210.9 27.9 466.0
NiCr steel (normal impact) 102.8 188.2 540 567.5
170 145.3 513.8
228.2 113.8 520.9
261 77 515.1
290.8 36.9 529.7
NiCr steel (low impact) 157.1 160.5 509 614.9
168.1 147.7 593.5
198.7 99.4 525.0
223.8 67.2 515.2
235.4 31.8 501.7
NiCrMo steel 104.2 194.8 594 675.6
169.8 147.6 602.5
237.3 123.8 645.4
252.9 75.5 589.3
265.7 36.6 574.5
Based on the data provided in Table 2, the agreement of the results in case of bending was expected because the size of the structural volume was determined for the fatigue properties of the material under bending loading. On
0.508 mm I \~J
0
3.81 mm
M
M
Fig. 6. Geometry of notched specimens with a sharp (a) and blunt notch (b)
the other hand, the limited error in the prediction of the fatigue limit under pure torsion and combined bending-torsion attests to the generalizability of the proposed model for different loading modes.
4.2. Application for tubular specimens with a sharp circumferential notch (2a = 60°)
The second set of data used to validate the method was gathered from Ref. [40]. The specimen geometries are shown in Fig. 6, a. In all cases, the notch depth was 0.508 mm and the notch-opening angle was 60°. The uniaxial fatigue properties, material characteristic length L, notch tip radius, and corresponding stress concentration factors are summarized in Table 4.
It should be noted that, regarding Eq. (11), the critical value of the notch stress intensity factor is required to calculate the value of Rc. However, there is no information about it for most of the considered materials. On the other hand, the material characteristic length L of the studied materials is provided in Ref. [40]. Since the value of Rc for the case of a cracked body (2a = 0°) and L are both proportional to (AKth/Act^ )2, the values of L were used here to calculate Rc. In ABAQUS, a circular sector with
Table 6
Accuracy in predicting fatigue limits of blunt V-notched specimens (2a = 90°) of SAE 1045 under bending and torsion (RL=-1)
a0, MPa T0, MPa Phase angle Fatigue limit, MPa oeff, MPa
188 0 - 287.8
0 150.7 - 328.3
157.2 44.5 0° 259.4
161.1 58.8 0° 304 277.9
150.2 82.6 0° 292.0
142.6 99.7 0° 307.9
94.2 134.9 0° 327.3
156.9 100.6 90° 325.1
P
Table 7
Accuracy in predicting fatigue limits of blunt V-notched specimens (2a = 90°) of S65 under bending and torsion with superimposed static stresses
ct0a, MPa* ctom, MPa* t0a, MPa* tom, MPa* Rl,ct Ca Rl,t Q cteff, MPa Error, %
0 0 270.2 0 -1 0.5 -1 0.5 527.1 -9.75
347.3 266.3 0 0 -0.1320 0.7940 -1 0.5 668.0 14.38
361.2 266.3 0 169.8 -0.1512 0.7718 1 1 684.9 17.28
318 266.3 0 169.8 -0.0885 0.85064 1 1 633.0 8.40
0 0 276.3 169.8 -1 0.5 -0.2387 0.6888 632.6 8.32
0 266.3 243.9 0 1 1 -1 0.5 475.8 -18.53
0 266.3 230 169.8 1 1 -0.1506 0.7725 557.7 -4.51
123.5 0 245.5 0 -1 0.5 -1 0.5 514.6 -11.88
255.4 0 169.8 0 -1 0.5 -1 0.5 511.5 -12.41
311.1 0 88 0 -1 0.5 -1 0.5 504.9 -13.55
312.6 266.3 208.4 0 -0.0800 0.8629 -1 0.5 747.0 27.92
294.2 0 196.1 169.8 -1 0.5000 -0.0719 0.8749 676.5 15.84
270.9 266.3 180.6 169.8 -0.0086 0.9832 -0.0308 0.9420 755.0 29.27
: Subscripts "a" and "m" refer to the amplitude and mean values of the stress component, respectively.
the radius R = 0.25 mm around the notch tip was used to determine the averaged value of the strain energy density. The effective stress values predicted within the proposed approach for this case under fully reversed loading are provided in Table 5. A good agreement can be found.
4.3. Application for tubular specimens with a blunt circumferential notch (2a = 90°)
The third set of data for method validation was gathered again from Ref. [40]. The geometry of the specimens is shown in Fig. 6, b. The notch opening angle in all cases is 90°, while the values of the notch tip radius and the corresponding stress concentration factors are provided in
Table 4. The value of Rc was again calculated according to the description given in the previous section based on the characteristic length L.
The effective stress values predicted within the proposed approach for this case model under fully reversed loading are presented in Tables 6 and 7. The tabulated data are given for out-of-the-phase loading, together with various load ratios. A good agreement can be seen.
The errors of the fatigue limit prediction based on the proposed approach for the cases investigated in Sects. 4.1 to 4.3, except for S65 specimens, are summarized in Fig. 7. The horizontal axis represents the ratio of the shear stress amplitude to the axial stress amplitude, and the vertical axis
100
20 0
0 <
< l
¡a
t> -20 < 20
o W
□ E 460 steel
O 0.4% C steel (normalized)
3% Ni steel O 3.5% Ni steel O Cr-Va steel
A 3.5% Ni-Cr steel (normal impact) > 3.5% Ni-Cr steel (low impact) < Ni-Cr-Mo steel * SAE 1045
-100
0 2 4 6
Load ratio = Acto/Ato
Fig. 7. Scatter band summarizing fatigue limit prediction error of notched components under multiaxial loading (color online)
represents the error index based on Eq. (18). The red lines in this graph represent a 20% error band. According to this graph, the proposed approach is acceptable for different materials, geometries, and relative intensities of multiaxial loads.
4. Conclusions
The strain energy density is one of the most fundamental quantities in mechanics, in terms of which various physical quantities can be expressed. In this paper, an approach based on the physical mechanism governing fatigue damage was proposed to predict the high cycle fatigue strength, i.e., fatigue limit, of notched components under multiaxial loading using the deviatoric strain energy density function as the driving force of failure. In this approach, regardless of the geometrical feature (smooth, sharp, or blunt notch), the averaged strain energy over a structural volume around the critical point of crack initiation is the basis for assessing notched components against multiaxial fatigue. New relationships for determining an effective (tensile-type) stress and fatigue strength reduction factors for notched specimens were developed.
The implementation steps of this criterion are as follows:
According to Eq. (11), the size of the structural volume is explicitly derived from the material high cycle fatigue properties, which can be obtained via simple uniaxial tests.
For a sharp notch, by using linear elastic finite element analysis, the averaged strain energy over a control volume of arbitrary radius is calculated. Finally, the fatigue strength reduction factors are calculated from Eqs. (14).
For a blunt notch, the parameters of Eqs. (15), (16) are determined with respect to the material and geometry of the notch. Finally, the fatigue strength reduction factors are calculated from Eqs. (17).
The effective stress applied to the structure under a specified loading condition is calculated via Eq. (121), which must be compared with the fatigue limit of the constituent material according to the critical conditions stated in Eq. (13).
The theoretical results obtained from the proposed approach were found to be in a good agreement with experimental data from the literature. Sharp and blunt notches as well as the effect of the load ratio were studied.
In general, regarding the findings of the references and the results of this paper, the advantages of using the strain energy density criterion in evaluating the fatigue failure of notched components can be summarized as follows:
By precisely defining the structural volume, the effect of microstructure on the process zone is fully addressed.
Evaluation of the strain energy density over the control volume is not sensitive to the degree of mesh refinement
Effect of different modes (I, II, III) is considered.
Effect of the load ratio with the appropriate coefficients is taken into account.
Under localized yielding conditions, the strain energy density is independent of the material response (linear elastic or elastic-plastic). Therefore, the proposed criterion coupled with linear elastic finite element analysis is capable of addressing the fatigue cracking process in the presence of plasticity at the notch tip, in particular for complex structures of practical interest.
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Received 17.09.2018, revised 02.10.2018, accepted 09.10.2018
Сведения об авторах
Mohammad Abolghasemzadeh, PhD Candidate, Amirkabir University of Technology, Iran, m.abolghasemzadeh@hotmail.com
Yoness Alizadeh, Assoc. Prof., Amirkabir University of Technology, Iran, alizadeh@aut.ac.ir
Hosein Mohammadi, MSc student, Amirkabir University of Technology, Iran, hosein.mohammadi@aut.ac.ir
